Final answer:
The polynomial with rational coefficients and zeros √2 and 3+1, and their conjugates, is found by multiplying the factors (x - √2)(x + √2)(x - (3+1))(x - (3-1)), resulting in the polynomial f(x) = x⁴ - 6x³ + 8x² + 12x - 20.
Step-by-step explanation:
To write a polynomial function f of least degree with rational coefficients, a leading coefficient of 1, and the given zeros √2, 3+1, we must consider the fact that the rational zeros theorem implies that if a polynomial has rational coefficients, any irrational or complex roots must come in conjugate pairs. Therefore, if √2 is a zero, its conjugate pair, -√2, must also be a zero. Similarly, if 3+1 is a zero, we must also include its conjugate, 3-1.
The polynomial we are looking for will have those four zeros: √2, -√2, 3+1, 3-1. We can represent these zeros as factors of the polynomial as follows: (x - √2)(x + √2)(x - (3+1))(x - (3-1)). Let's first focus on the factors with square roots:
(x - √2)(x + √2) = x² - 2,
since it is the difference of squares. Next, we look at the complex factors and multiply them:
(x - (3+1))(x - (3-1)) = (x - 4)(x - 2) = x² - 6x + 8.
Now we can find f(x) by multiplying these two results:
f(x) = (x² - 2)(x² - 6x + 8)
Multiplying these quadratic expressions will give us the polynomial of least degree that satisfies the conditions, which results in:
f(x) = x⁴ - 6x³ + 8x² + 12x - 20.
This polynomial meets all the criteria given, and therefore the correct answer is f(x) = x⁴ - 6x³ + 8x² + 12x - 20.